= Rapidity
{title2=$\theta=\operatorname{artanh}v$}
{wiki}
<Rapidity> is a dimensionless parameter for a signed collinear <velocity> $u$ with $|u|<c$, defined using the <inverse hyperbolic functions> by
$$
\theta=\operatorname{artanh}(u/c)=\frac12\ln\frac{1+u/c}{1-u/c},\qquad u=c\tanh\theta.
$$
Here $c$ is the <speed of light>, and $\tanh$ is the <hyperbolic tangent>. Zero <velocity> has zero <rapidity>; negative <velocity> has negative <rapidity>. Every finite real <rapidity> gives $|u|<c$, while $u\to\pm c$ requires $\theta\to\pm\infty$. Thus a <light> ray has no finite <rapidity> in this parametrization. The header's formula uses <natural units> $c=1$: its $v$ is the numerical <velocity> $u/c$.
For the <inertial frames> in the <velocity-addition formula>, $S'$ moves at signed <velocity> $V$ along the $x$-axis of $S$. Set $\phi=\operatorname{artanh}(V/c)$. Re-expressing the same particle in the moving <inertial frame> gives
$$
\theta'=\theta-\phi,\qquad \theta=\theta'+\phi.
$$
The subtraction is a passive change of <reference frame>; the inverse adds the particle's <rapidity> in $S'$ to the frame's <rapidity> relative to $S$. For example, $V=0.6c$ and $u'=0.6c$ give $\phi=\theta'=\ln2$, so $\theta=\ln4$ and $u=c\tanh(\ln4)=15c/17$, exactly as in the <velocity-addition formula>.
For a massive particle of <rest mass> $m>0$ moving in one spatial dimension, its <energy> and signed <momentum> are
$$
E=mc^2\cosh\theta,\qquad p=mc\sinh\theta.
$$
The <hyperbolic cosine> is the particle's <Lorentz factor>, and the <hyperbolic sine> is $\gamma u/c$. The <hyperbolic Pythagorean identity> immediately gives the <relativistic energy-momentum relation> $E^2-c^2p^2=m^2c^4$. In the older $c=1$ notation, writing the <rest mass> as $M$ recovers $v=\tanh\theta$, $E=M\cosh\theta$, and $p=M\sinh\theta$. These expressions use <natural units> in which the factors of $c$ are numerically one; the preceding formulas restore those factors explicitly. https://davidtong.org/pdfs/teaching/dynamics-and-relativity/dynrel7.pdf[David Tong, Dynamics and Relativity, §7.3.2: rapidity; §7.4.3: energy and momentum]{ref}.
The useful simplification follows from the <hyperbolic-function identity> $\tanh(a+b)=(\tanh a+\tanh b)/(1+\tanh a\tanh b)$: the nonlinear <velocity-addition formula> becomes ordinary addition of <rapidities>. See <iterated collinear boosts with equal rapidity> for a plotted example. A common collinear <Lorentz boost> shifts every <rapidity> by the same constant, so a <rapidity> difference is invariant. This makes <rapidity> useful for relativistic <factorized scattering> in one spatial dimension, where two-body <S-matrices> depend on the difference of the incoming <rapidities>, with particle species fixed.
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